Abstract
This paper introduces the concept of fuzzy projection of a fuzzy number on a set of fuzzy numbers based on r-cut approach. It is proved that the projection of a fuzzy number on the set of all fuzzy numbers is itself and under a special metric, the proposed fuzzy projection is a non-expansive mapping. By using this definition, the concept of fuzzy linear projection equation is defined and to solve it, a numerical method is applied. Based on the proposed algorithm and as an important application, two different types of system of fuzzy linear equations with fuzzy variables are solved. Numerical results illustrate the applicabilities of proposed approach.
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References
Amemiya, M., & Takahashi, W. (2000). Generalization of shadows and fixed point theorems for fuzzy sets. Fuzzy Sets and Systems, 114, 469–476.
Bede, B. (2013). Mathematics of fuzzy sets and fuzzy logic. Berlin: Springer.
Behera, D., & Chakraverty, S. (2013). Solution to fuzzy system of linear equations with crisp coefficients. Fuzzy Information and Engineering, 5, 205–219.
Bernstein, D. S. (2009). Matrix mathematics, theory, facts, and formulas. Princeton: Princeton University Press.
Chang, S., & Zhu, Y. (1989). On variational inequalities for fuzzy mappings. Fuzzy Sets and Systems, 32, 359–368.
Crouzet, J. F. (2012). Fuzzy projection versus inverse fuzzy transform as sampling/interpolation schemes. Fuzzy Sets and Systems, 193, 108–121.
Dehghan, M., & Hashemi, B. (2006). Iterative solution of fuzzy linear systems. Applied Mathematics and Computation, 175, 645–674.
Fang, S.-C., & Hu, C.-F. (2002). Solving fuzzy variational inequalities. Fuzzy Optimization and Decision Making, 1, 113–133.
Friedman, M., Ming, M., & Kandel, A. (1998). Fuzzy linear systems. Fuzzy Sets and Systems, 96, 201–209.
Gupta, S. K., & Dangar, D. (2014). Duality for a class of fuzzy nonlinear optimization problem under generalized convexity. Fuzzy Optimization and Decision Making, 13, 131–150.
Kinderlehrer, D., & Stampacchia, G. (1980). An introduction to variational inequalities and their applications. In S. Eilenberg and H. Bass (Eds.), SIAM classics in applied mathematics. London: Academic Press.
Korpelevich, G. M. (1976). The extragradient method for finding saddle points and other problems. Ekonomicheskikh Matematicheskie Metody, 12, 747–756.
Nemeth, A. B., & Nemeth, S. Z. (2013). Lattice-like operations and isotone projection sets. Linear Algebra and its Applications, 439, 2815–2828.
Perfilieva, I. (2004). Fuzzy transforms. In Transactions on Rough Sets II, Lecture Notes in Computer Science (Vol. 3135, pp. 63–81).
Roman, H. F., & Flores, A. F. (2001). A note on projection of fuzzy sets on hyperplanes. Proyecciones, 20, 339–349.
Rufian-Lizanaa, A., Chalco-Canob, Y., Osuna-Gomeza, R., & Ruiz-Garzonc, G. (2012). On invexf uzzy mappings and fuzzy variational-like inequalities. Fuzzy Sets and Systems, 200, 84–98.
Slavakis, K., Theodoridis, S., & Yamada, I. (2008). Online kernel-based classification using adaptive projection algorithms. IEEE Transections on Signal Processing, 56, 2781–2796.
Stefanini, L. (2010). A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets and Systems, 161, 1564–1584.
Theodoridis, S., Slavakis, K., & Yamada, I. (2011). Adaptive learning in a world of projections. IEEE Signal Processing Magazine, 28, 97–123.
Wang, X., Zhong, Z., & Ha, M. (2001). Iteration algorithms for solving a system of fuzzy linear equations. Fuzzy Sets and Systems, 119, 121–128.
Wu, H.-C. (2004). Duality theory in fuzzy optimization problems. Fuzzy Optimization and Decision Making, 3, 345–365.
Wu, Z., & Xu, J. (2009). Generalized convex fuzzy mappings and fuzzy variational-like inequality. Fuzzy Sets and Systems, 160, 1590–1619.
Zhang, H., Huang, B., Gong, D., & Wang, Z. (2013). New results for neutral-type delayed projection neural network to solve linear variational inequalities. Neural Computing and Applications, 23, 1753–1761.
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The authors are thankful to the unknown referees for their valuable and informative comments that have significantly improved the quality of the paper. They also appreciate the Editors and Editor in Chief for their valuable suggestions.
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Pakdaman, M., Effati, S. On fuzzy linear projection equation and applications. Fuzzy Optim Decis Making 15, 219–236 (2016). https://doi.org/10.1007/s10700-015-9222-8
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DOI: https://doi.org/10.1007/s10700-015-9222-8